Rank the following in order of increasing moments of inertia about the midpoint of the rod. m R 1 m 2m R/2 2 m m/2 2R 3 m/2

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**Ranking Moments of Inertia**

The task is to rank the following systems in order of increasing moments of inertia about the midpoint of the rod:

1. **Diagram 1:**
   - **Masses:** Two identical masses \( m \)
   - **Distance:** Each mass is located at a distance \( R \) from the midpoint.

2. **Diagram 2:**
   - **Masses:** A mass \( 2m \) and a mass \( m \)
   - **Distance:** The mass \( 2m \) is at a distance \( R/2 \) from the midpoint, and the mass \( m \) is at the same distance \( R/2 \) on the opposite side.

3. **Diagram 3:**
   - **Masses:** Two identical masses \( m/2 \)
   - **Distance:** Each mass is located at a distance \( 2R \) from the midpoint.

**Explanation:**

- **Moment of Inertia (I)** is calculated using the formula \( I = \sum m_i r_i^2 \), where \( m_i \) is the mass and \( r_i \) is the distance from the axis of rotation (midpoint in this case).
  
For each system:
- **Diagram 1:** Moment of inertia is \( 2mR^2 \).
- **Diagram 2:** Moment of inertia is \( (2m)(R/2)^2 + (m)(R/2)^2 = \frac{3}{2}m \left( \frac{R^2}{4} \right) = \frac{3}{4}mR^2 \).
- **Diagram 3:** Moment of inertia is \( 2 \times (m/2) (2R)^2 = 8mR^2 \).

**Rank:**
2 (Smallest) < 1 < 3 (Largest)
Transcribed Image Text:**Ranking Moments of Inertia** The task is to rank the following systems in order of increasing moments of inertia about the midpoint of the rod: 1. **Diagram 1:** - **Masses:** Two identical masses \( m \) - **Distance:** Each mass is located at a distance \( R \) from the midpoint. 2. **Diagram 2:** - **Masses:** A mass \( 2m \) and a mass \( m \) - **Distance:** The mass \( 2m \) is at a distance \( R/2 \) from the midpoint, and the mass \( m \) is at the same distance \( R/2 \) on the opposite side. 3. **Diagram 3:** - **Masses:** Two identical masses \( m/2 \) - **Distance:** Each mass is located at a distance \( 2R \) from the midpoint. **Explanation:** - **Moment of Inertia (I)** is calculated using the formula \( I = \sum m_i r_i^2 \), where \( m_i \) is the mass and \( r_i \) is the distance from the axis of rotation (midpoint in this case). For each system: - **Diagram 1:** Moment of inertia is \( 2mR^2 \). - **Diagram 2:** Moment of inertia is \( (2m)(R/2)^2 + (m)(R/2)^2 = \frac{3}{2}m \left( \frac{R^2}{4} \right) = \frac{3}{4}mR^2 \). - **Diagram 3:** Moment of inertia is \( 2 \times (m/2) (2R)^2 = 8mR^2 \). **Rank:** 2 (Smallest) < 1 < 3 (Largest)
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